Laws of exponents Problem set B Worked solutions

Write \((2x^3)^4 \times 5x^{-9}\) in the form \(ax^p\).

• \((2x^3)^4 \times 5x^{-9} = 16x^{12} \times 5x^{-9}\)
• Multiply the numbers, add the powers of x.
• \(80x^{3}\)

Find the exact value of:

1. (a) \(3^{-4}\).

   • \(3^{-4} = \frac{1}{3^{4}}\)
   • \(= \frac{1}{81}\)

2. (b) \(\left(\frac{2}{5}\right)^{-3}\).

   • \(\left(\frac{2}{5}\right)^{-3} = \left(\frac{5}{2}\right)^{3}\)
   • \(= \frac{125}{8}\)

Write \(\frac{(4x^3)^2}{8x^{-4}}\) in the form \(ax^p\).

• \((4x^3)^2 = 16x^{6}\)
• Divide the numbers and subtract the powers: \(16 \div 8\) and \(6 - (-4) = 10\).
• \(2x^{10}\)

Find the exact value of:

1. (a) \(5^{-3}\);

   • \(5^{-3} = \frac{1}{5^3}\)
   • \(= \frac{1}{125}\)

2. (b) \(\left(\frac{3}{2}\right)^{-3}\);

   • \(\left(\frac{3}{2}\right)^{-3} = \left(\frac{2}{3}\right)^{3}\)
   • \(= \frac{8}{27}\)

3. (c) \(7^0 - 2^{-2}\).

   • \(7^0 = 1\) and \(2^{-2} = \frac{1}{4}\)
   • \(1 - \frac{1}{4} = \frac{3}{4}\)
   • \(= \frac{3}{4}\)

Write each answer in the form \(a\times10^{k}\).

1. (a) Write 0.000 061 3 in standard form.

   • Move the point 5 places right to make a = 6.13
   • \(0.000\,061\,3=6.13\times10^{-5}\)

2. (b) Find \((3.5\times10^{7})\times(6\times10^{-10})\).

   • \(3.5\times6=21\) and \(10^{7}\times10^{-10}=10^{-3}\)
   • \(21\times10^{-3}=2.1\times10^{-2}\)

3. (c) Find \((8.4\times10^{7})+(3\times10^{8})\).

   • \(8.4\times10^{7}=0.84\times10^{8}\)
   • \(0.84\times10^{8}+3\times10^{8}=3.84\times10^{8}\)

4. (d) Find \(\dfrac{1.2\times10^{-2}}{4\times10^{-9}}\).

   • \(1.2\div4=0.3\) and \(10^{-2}\div10^{-9}=10^{7}\)
   • \(0.3\times10^{7}=3\times10^{6}\)

The strength of a Wi-Fi signal is \(S = 54d^{-2}\) units at a distance of d metres from the router. Find the exact strength at a distance of:

1. (a) \(d = \frac{3}{2}\) m;

   • \(54 \times \left(\frac{3}{2}\right)^{-2} = 54 \times \left(\frac{2}{3}\right)^{2}\)
   • \(= 24\)

2. (b) \(d = 6\) m.

   • \(54 \times 6^{-2} = \frac{54}{36}\)
   • \(= \frac{3}{2}\)

The area of mould on a wall, in cm², is \(M = 5 \times 3^t\), where t is the time in days after it was first measured. A negative t means before it was first measured. Find the exact area:

1. (a) 3 days after it was measured;

   • \(M = 5 \times 3^{3} = 5 \times 27\)
   • \(= 135\)

2. (b) 2 days before it was measured.

   • \(M = 5 \times 3^{-2} = 5 \times \frac{1}{9}\)
   • \(= \frac{5}{9}\)

One drive in a data centre stores \(4.2\times10^{12}\) bytes and a second drive stores \(8.5\times10^{11}\) bytes.

1. (a) Find the total number of bytes stored, in the form \(a\times10^{k}\).

   • \(8.5\times10^{11}=0.85\times10^{12}\)
   • \(4.2\times10^{12}+0.85\times10^{12}=5.05\times10^{12}\) bytes

2. (b) How many times more does the first drive store than the second? Give your answer to 3 significant figures.

   • \(\dfrac{4.2\times10^{12}}{8.5\times10^{11}}\)
   • From the calculator: 4.94

Write \(\frac{(2x^{-3})^3 \times 9x^7}{12x^{-5}}\) in the form \(ax^p\).

• \((2x^{-3})^3 = 8x^{-9}\)
• Numerator: \(8 \times 9 = 72\) and \(x^{-9} \times x^7 = x^{-2}\).
• Divide by \(12x^{-5}\): the number is \(72 \div 12\) and the power is \(-2 - (-5) = 3\).
• \(6x^{3}\)